The random variable X can take values 0, 1, 2. If $P(X=0) = P(X=1)=α$, and $E(X^2)=E(X)$, then which of the following are correct?

(A) $E(X) = 2-3α$
(B) $E(X^2)= 4+7α$
(C) $α =\frac{1}{2}$
(D) $α =\frac{1}{5}$

Choose the correct answer from the options given below:

(A) and (C) only

The correct answer is Option (4) → (A) and (C) only

$X \in \{0,1,2\}.$

$P(X=0)=P(X=1)=\alpha.$

$P(X=2)=1-2\alpha.$

$E(X)=0\cdot\alpha+1\cdot\alpha+2(1-2\alpha).$

$E(X)=\alpha+2-4\alpha=2-3\alpha.$

$\text{Thus (A) is correct.}$

$E(X^2)=0^2\alpha+1^2\alpha+2^2(1-2\alpha).$

$E(X^2)=\alpha+4-8\alpha=4-7\alpha.$

$E(X^2)=E(X).$

$4-7\alpha=2-3\alpha.$

$2=4\alpha.$

$\alpha=\frac{1}{2}.$

$\text{Thus (C) is correct.}$

$\text{(B) is incorrect since }E(X^2)=4-7\alpha.$

$\text{(D) is incorrect.}$

$\text{Correct options: (A) and (C).}$